398 
PROFESSOR J. CASEY ON A NEW 
hence its first positive pedal is 
a sin 2 p 
^ cos<p ’ 
( 144 ) 
which equation represents, as is well known, the cissoid. 
Again, the pedal of the logarithmic curve is 
g=«sin <p log tan <p, (145) 
and of the ellipse 
q—sJ a? sin 2 cos 2 <p. . . (146) 
This curve is a bicircular quartic. 
53. The tangential equation of the evolute of the curve v—f{$) is 
P=f(<P) cot <p-}-/'(<p). 
Hence the polar equation of the first positive pedal of the evolute is 
g =/(<?) cos ? +/'( < P) sin< P (147) 
54. The foregoing result can be shown geometrically as follows (see fig. art. 26) : — 
The perpendicular OT on PQ is the radius vector of the pedal of the evolute ; hut 
OT=OL cos <p-J-LP 
=/(<P) cos <p+/'(<p) sin <p. 
Cor. 1. The equation (147) may be written 
?=^(/( < P) sin< P) (1-48) 
This also appears from art. 43 ; and from the same article we see that the first positive 
pedal of the nth evolute is 
S»=(^) (/fa) sin< P) (149) 
Cor. 2. If in the last equation n be taken as negative, we have the first positive pedal 
of the nth involute. 
Cor. 3. If g, and denote the radii vectores of the first positive pedals of the evolute 
and involute of v= /(<p), we have 
f* =fW sin c P+/( < P) cos 
§- 1 = -/(<p) cos <p+J/'(<p) cos <p dcp ; 
• •• ?i+f-i=/'(<P) sin <p +J/'(<p) cos <p d<p, 
••• ^+^=s : (150) 
H^nce we have the following theorem : — 
The length of a curve is equal to the sum of the radii vectores of the first positive 
pedals of its evolute and involute. 
